Summary: This article gives theoretical insights into the performance of K-SVD, a dictionary learning algorithm that has gained significant popularity in practical applications. The particular question studied here is when a dictionary \(\varPhi\in\mathbb R^{d\times K}\) can be recovered as local minimum of the minimisation criterion underlying K-SVD from a set of \(N\) training signals \(y_n=\varPhi x_n\). A theoretical analysis of the problem leads to two types of identifiability results assuming the training signals are generated from a tight frame with coefficients drawn from a random symmetric distribution. First, asymptotic results showing that in expectation the generating dictionary can be recovered exactly as a local minimum of the K-SVD criterion if the coefficient distribution exhibits sufficient decay. Second, based on the asymptotic results it is demonstrated that given a finite number of training samples \(N\), such that \(N/\log N=O(K^3 d)\), except with probability \(O(N^{-Kd})\) there is a local minimum of the K-SVD criterion within distance \(O(K N^{-1/4})\) to the generating dictionary.